
We remind the readers that \([j,z|\)  is called the dual spinor due to the fact that it is orthogonal to the regular spinor \(|j, z\rangle \) , \(\textit {i.e.}\)  \([j,z|j,z\rangle =0\) , but not because it lives in the dual representation \({\mathcal {V}}^{j*}\) . \(|j,z]=|j,\varsigma z\rangle \)  is the dual spinor living in \({\mathcal {V}}^j\)  and anti-holomorphic on \(z^A\) .

Spinors have been used to construct the spinorial phase space of loop gravity, which is equivalent to the holonomy-flux phase space [1]}. A brief summary is given in Appendix .
